Mathematica Subroutine
(Runge-Kutta
Method) To compute
a numerical approximation for the solution of the initial value
problem with over
the interval at
a discrete set of points using the formula

,

where , , ,
and .

Example
1.Solve the
I.V.P. over .Use
the Runge-Kutta method.Solution
1.

Extension to 2D. The
Runge-Kutta method is easily extended to solve a system of D.E.'s
over the
interval .

Mathematica Subroutine
(Runge-Kutta
Method in 2D space) To
compute a numerical approximation for the solution of the initial
value problem

with , with ,

over the interval at
a discrete set of points.

Note. The
Runge-Kutta method in 2D is a "vector form" of the one-dimensional
method, here the function f is
replaced with F.

Example
2. Lotka-Volterra Model. Solve the
I.V.P. with ,
andwith .
Use several intervals .2 (a). Use the
interval .
2 (b). Use the
interval .
2 (c). Use the
interval .
Can you discover if the solution form an "orbit."Solution
2.

Example
3. Lotka-Volterra Model. Solve the
I.V.P. with ,
andwith .
Combine the system of D. E.'s to form a separable first-order
differential equation and solve the D. E..Solution
3.

Example 4. For
the I.V.P. with ,
andwith .
Show that the numerical solution in Example 3 and the analytic
solution in Example 4 are in agreement.Solution
4.

Example 5. For
the I.V.P. with ,
andwith .
The implicit solution is .
Determine if y can be solved as a function of x.Solution
5.

Predator-Prey Model

The study of population dynamics of competing
species is attributed two two independently published works by
Alfred
James Lotka (1880 - 1949) and Vito
Volterra (1860-1940).

Consider two two species, the predator is population is y(t)
(foxes), and the prey population is x(t)
(rabbits). It is assumed that the prey, x(t),
has adequate food and ,
are the birth rate and death rates, respectively. An
additional term, , contributing
to the decrease of prey is due to successful hunting of the
predators. Combining these 3 quantities, we obtain the
rate of change of

.

The substitution ,
will help simplify this result, and we obtain

.

The birth rate for the predator is proportional to its food supply,
x(t), i.e.the birth rate (predators)
is , and
the death rate of the predators is ,
and we obtain

.

These two equations are an application of the Lotka-Volterra
equations.

Example 6. Assume
that the initial number of foxes and rabbits
are
and ,
respectively, and that the
coefficients , are
used to form the system of D. E.'s
and Solve the system of D. E.'s for
x(t)and
x(t)
over the interval . Solution
6.